Some Remarks of the Kollar and Mori's Birational Geometry of Algebraic Varieties I

This blog aim to give some remarks and complete some details in this book (Kollar and Mori’s Birational Geometry of Algebraic Varieties). I will read first five chapters of this book. This is the first blog from chapter 1 to chapter 2.

Chapter 1. Rational Curves and the Canonical Class

Section 1.1. Finding Rational Curves when $K_X$ is Negative

Theorem 1.10. For the deformation space of maps $f:C\to X$, if we consider the deformation space which fixed points $p_1,…,p_k$, then one can show that \[\dim(\mathrm{DefSpace})\geq\chi(C,f^* T_X\otimes\mathscr{I}_{p_1+…+p_k}).\] If $C$ smooth (at least at $p_i$), then by Tag 0AYV, we get

\begin{align*} &\dim(\mathrm{DefSpace})\geq\chi(C,f^* T_X\otimes\mathscr{O}(-p_1-...-p_k))\\ &=\chi(C,f^* T_X)-k\dim X\\ &=-(f_* (C)\cdot K_X)+(1-g(C)-k)\dim X \end{align*}

and well done. $\blacksquare$

Section 1.3. The Cone of Curves of Smooth Varieties

Definition 1.15. More properties of extremal faces and rays we refer chapter 18 (especially Theorem 18.5) in book [Convex97]¹ which is important for us to read the Mori’s theory. $\blacksquare$

Theorem 1.24. At the last step, in the process of proving $\overline{NE}(X)_{K_X\geq 0}+\sum_i\mathbb{R}_{\geq 0}[C_i]$ is closed, one can get that these $\mathbb{R}_{\geq 0}[C_i]$ are extremal rays. $\blacksquare$

Section 1.4. Minimal Models of Surfaces

Theorem 1.28. The second paragraph of the case $C^2=0$. Easy to see that we must have $k[C]=\sum a_i[C_i]=:F$ in $R\subset \overline{NE}(X)$ where $k\neq 0$ by checking intersection numbers. As $Z$ be a smooth curve, we get $\mathrm{cont}_R:X\to Z$ is flat (as over Dedekind domain, flat iff torsion-free). Hence we have $k=1$ and hence $[C]=\sum a_i[C_i]$!

For more methods, let $\sum a_i[C_i]=:F$. As $R$ be a extremal ray, we get $[C_i]\in R$ for all $i$. We find that $a_iC_i^2=C_i\cdot (F-\sum_{j\neq i}C_j)$. As $C_i\cdot F=0$, we get $a_iC_i^2=-\sum_{j\neq i}C_i\cdot C_j\leq 0$. If $F$ is reducible, then $C_i^2<0$ since the fibers are connected. As $c_i^2=0$, we get $F$ is irreducible. Hence we also can get $F=aD$ for an integral curve $D$. $\blacksquare$

Chapter 2. Introduction to the Minimal Model Program

Section 2.3. Singularities in the Minimal Model Program

Definition 2.25. Here they just defined $a(E,X,\Delta)$ for some irreducible exceptional divisor over $X$ (that is, $\mathrm{codim}_X(\mathrm{center}_XE)\geq 2$). But actually this book told us that by definition we have $a(D_i,X,\Delta)=-a_i$ and $a(D,X,\Delta)=0$ for these non-exceptional divisor! Although these claim are very intuitive, we cannot do this AT ALL! Actually, following Definition 2.4 in the new book [SMMP]² (also by Kollar!), we need to DEFINE $a(D,X,\Delta):=-\mathrm{coeff}_D \Delta$ for any non-exceptional divisor $D\subset X$! $\blacksquare$

Lemma 2.27. In fact, by definition one can see that \[a(E,X,\Delta)=a(E,X,\Delta+\Delta’)+\mathrm{coeff}_E f^* \Delta’\] and well done. $\blacksquare$

Lemma 2.29. Let $Y:=\mathrm{Bl}_Z X$ with $p:Y\to X$. We first let $Z$ is a smooth subvariety of codimension $k$ in $X$, then by Exercise II.8.5(b) in [Har77]³ we get \[K_Y=p^* K_X+(k-1)E.\] Hence we get \[p^* (K_X+\Delta)=K_Y+p^{-1}_* \Delta+(1-k+\mathrm{mult}_Z\Delta)E\] and we have $a(E,X,\Delta)=k-1-\mathrm{mult}_Z\Delta$ and well done.

When $Z\subset X$ just a subvariety of codimension $k$, as for now $E$ be a component of the exception divisor which dominates $Z$, then we can replace $X$ with any open subscheme that contains the generic point of $Z$. Hence we can replace $X$ by $X\backslash\mathrm{Sing}(Z)$ and back to the smooth case! $\blacksquare$

Corollary 2.31. (1) Here fixed a birational $f:Y\to X$, we let \[\Delta_Y=f_* ^{-1}\Delta-\sum_{E_i,f-\mathrm{exceptional}}a(E_i,X,\Delta)E_i\] Pick $Z_0$ of codim $2$ subvariety in $E$ which not in any other $f$-exceptional divisors and not in $f^{-1}_* \Delta$. Hence for $E_k\subset Y_k$, inductively we get

\begin{align*} &a(E_k,Y,\Delta_Y)=1-\mathrm{mult}_{Z_{k-1}}\Delta_Y\\ &=1-\mathrm{mult}_{Z_0}\left(f_*^{-1}\Delta-\sum_{E_i,f-\mathrm{exceptional}}a(E_i,X,\Delta)E_i\right)\\ &=1-\mathrm{mult}_{Z_0}\left(-a(E_{k-1},X,\Delta)E_{k-1}-a(E,X,\Delta)E\right)\\ &=1+(1-k)c+(-1-c)=-kc \end{align*}

and well done. $\blacksquare$

(3) Where the snc divisor we used? First: As $\sum_i D_i$ is snc, then for $J\subset I$ finite the scheme theoretic intersection $D_J:=\bigcap_{j\in J}D_j$ is a regular scheme each of whose irreducible components has codimension $\sharp(J)$ in $X$. Hence for the situation now, $\mathrm{codim} f(D)=k$ and we can let $f(D)\subset D_i$ iff $i\leq b$ for some $b\leq k$.

Second: Actually the multiplicity $\mathrm{mult}_YD$ of a divisor $D$ along a subvariety $Y\subset X$ can be defined as the coefficient of exceptional divisor in the pullback of $D$ along $\mathrm{Bl}_Y X\to X$. This definition can deduce from the definition ($\mathrm{coeff}_{[Y]}s(Y,D)$) in section 4.3 of book [FulIT2nd]⁴ (see also section 5.2.B in book [Positivity-I]⁵). Hence for the situation now, if $\Delta=\sum_i a_iD_i$ such that $\sum_i D_i$ is snc, then for some subvariety $Z$ we easy to see that \[\mathrm{mult}_Z\Delta=\sum_{Z\subset D_i}\mathrm{coeff}_{D_i}\Delta\] and this is the reason of snc divisors! $\blacksquare$

Corollary 2.35.(3). As we have \[a(E,X,\Delta)=a(E,X,\Delta+\varepsilon\Delta’)+\varepsilon\mathrm{coeff}_E f^* \Delta’\] and we may need not assume that $\Delta,\Delta’$ have no common irreducible components (I think). $\blacksquare$

Proposition 2.36.(2) Finally, we get \[\mathrm{totaldiscrep}(X,\Delta)=-\max_i(1-\alpha_i)\] by definition. Hence $1+\mathrm{totaldiscrep}(X,\Delta)=\min_i(\alpha_i)$ and well done. $\blacksquare$

Proposition 2.43.(Important definitions and facts). Here we need to introduce more things about Weil divisors and we follows [SMMP]² and the stacks project and a note [GD]⁶.

(I) Reflexive modules. Let $X$ be an integral locally Noetherian scheme and $F$ be a coherent $\mathscr{O}_X$-module. The reflexive hull of $F$ defined as $F^{\vee\vee}:=(F^{\vee})^{\vee}$. We called $F$ is reflexive if the natural map $j:F\to F^{\vee\vee}$ is an isomorphism (more results about these we refer St 31.12). We may define $L^{[m]}:=(L^{\otimes m})^{\vee\vee}$. By Tag 0EBL we find that if $X$ be an integral locally Noetherian normal scheme and $F,G$ are coherent reflexive $\mathscr{O}_X$-modules. Then $(F^{\vee}\otimes G)^{\vee\vee}\cong\mathscr{H}om(F,G)$ which shows that the rule $(F,G)\mapsto F\hat{\otimes}G:=(F\otimes G)^{\vee\vee}$ defines an abelian group law on the set of isomorphism classes of rank $1$ coherent reflexive $\mathscr{O}_X$-modules.

(II) Weil divisors and reflexive modules. Fixed an integral locally Noetherian normal scheme $X$ and consider the Weil divisor class group $\mathrm{Cl}(X)$ and the group of isomorphism classes of rank $1$ coherent reflexive $\mathscr{O}_X$-modules $\mathrm{Cl}’(X)$. First we define the map $\mathrm{Cl}(X)\to \mathrm{Cl}’(X)$ as $D\mapsto\mathscr{O}_X(D)$ as: since $X$ is normal, we can find some open set $U$ such that $U$ is regular and $\mathrm{codim}_X X\backslash U\geq 2$. Hence $\mathrm{Cl}(X)\cong \mathrm{Cl}(U)$ canonically and induce $\mathscr{O}_U(D_U)$. By the uniquely extension of reflexive $\mathscr{O}_X$-modules Tag 0EBJ, we get this map! Actually a more natural definition is that for any open $V\subset X$ we define \[\Gamma(V,\mathscr{O}_X(D)):=\{f\in K(X):\mathrm{div}(f)|_V + D|_V \geq 0\}.\] Second, by Tag 0EBM, its not hard to show that this map is an isomorphism!

(III) Global sections. Fixed an integral locally Noetherian normal scheme $X$ and a Weil divisor $D$ on it. Pick a non-zero $s\in\Gamma(X,\mathscr{O}_X(D))$. One can trivially construct a effective divisor $(s)_0=_{\mathrm{lin}} D$ and by Proposition 3.12 in [GD]⁶, we have many results as the non-singular case! $\blacksquare$

Theorem 2.44. Since the original paper [Sza95]⁷ has expired, we give a new reference about this. We refer book [Fujino17]⁸ Proposition 2.3.20. $\blacksquare$

Section 2.4. The Kodaira Vanishing Theorem

Definition 2.49. Now $X_{m,L}:=\underline{\mathrm{Spec}}_X\bigoplus_{i=0}^{m-1}L^{-i}$ with canonical map $p:X_{m,L}\to X$. Now we will find the kernel of $p^* :\mathrm{Pic}(X)\to\mathrm{Pic}(X_{m,L})$! Easy to see that $p_* \mathscr{O}_{X_{m,L}}=\bigoplus_{i=0}^{m-1}L^{-i}$. Let $M\in\ker p^* $, then we get \[p_* \mathscr{O}_{X_{m,L}}=p_* p^* M=M\otimes p_* \mathscr{O}_{X_{m,L}}=\bigoplus_{i=0}^{m-1}M\otimes L^{-i}.\] By the Krull-Schmidt theorem, the decomposition of a vector bundle in a direct sum of indecomposable ones is unique up to permutation of the summands, so we get $M\cong L^i$ for any $i$. Hence $\ker p^* =\{L^i\}\cong\mathbb{Z}/m\mathbb{Z}$. $\blacksquare$

All of the section 2.4. I will omit the whole proof of the Kodaira vanishing theorem here (since it is so old I think…). Next time I will read the section in the Fujino’s [Fujino17]⁸ using the mixed Hodge structure. (ps: now you used the GAGA, why we not using the kahler identity and kill these things?) $\blacksquare$

Section 2.5. Generalizations of the Kodaira Vanishing Theorem

Definition 2.59. Let $X$ be a normal projective variety with Cartier divisor $D$ and let $N(D)=\{m\geq 0:h^0(X,mD)\neq 0\}$. We may define

$$\kappa(X,D):=\left\{ \begin{align*} \max_{m\in N(D)}\dim\phi_{|mD|}(X),&N(L)\neq\emptyset\\ -\infty,&N(L)=\emptyset \end{align*}\right. $$

be the Iitaka dimension of $D$. When $X$ is not normal, we may take its normalization $\nu:X’\to X$ and define $\kappa(X,D):=\kappa(X’,\nu^* D)$.

So we can let $D$ is big if $\kappa(X,D)=\dim X$ which is equivalent to the definition here (see Lemma 2.2.3 in [Positivity-I]⁵). (Of course we can define the Iitaka dimension using $\mathrm{trdeg}_{\mathbb{C}}(R(X,D))-1$ where $R(X,D):=\bigoplus_{m\geq 0}H^0(X,mD)$.) We also define the volume of the big divisor $D$ to be \[\mathrm{vol}(X,D):=\limsup_{m\to +\infty}\frac{h^0(X,mD)}{m^{\dim X}/(\dim X)!}\] and by [Positivity-I]⁵ we get $\mathrm{vol}(X,D)=\lim\limits_{m\to +\infty}\frac{h^0(X,mD)}{m^{\dim X}/(\dim X)!}$ (for the case not big, it is not known that if this limit exists). $\blacksquare$

Theorem 2.64. For the final step, we have showed that $H^i(X,L\otimes f_* \omega_Y)=0$ for $i>0$. We may ask in this situation if we have $f_* \omega_Y=\omega_X$? Actually we have more general results (taken from Lemma 3.3.2 in [Fujino17]⁸):

Lemma. Let $X$ be a smooth variety and let $D$ be an $\mathbb{Q}$-divisor on $X$ such that $\mathrm{supp}(\{D\})$ is a snc divisor. If $Y$ be a log resolution of pair $(X,\{D\})$, then we have \[f_* \mathscr{O}_Y(K_Y+\lceil f^* D \rceil)\cong \mathscr{O}_X(K_X+\lceil D \rceil).\]

Sketch of the proof. Let $\Delta:=\lceil D \rceil-\lfloor D \rfloor$, then $\Delta$ is reduced snc divisor. Hence we get \[K_Y+f^{-1}_* \Delta+f^* \lfloor D \rfloor=f^* (K_X+\lceil D \rceil)+\sum_{E_i:f-\mathrm{exceptional}}a(E_i,X,\Delta)E_i.\] We can easily check that \[\mathrm{mult}_{E_i}(\lceil f^* D \rceil-(f^{-1}_* \Delta+f^* \lfloor D \rfloor))\geq 1\] for any $f$-exceptional with $a(E_i,X,\Delta)=-1$. Hence we have \[K_Y+\lceil f^* D \rceil=f^* (K_X+\lceil D \rceil)+F\] for some effective $f$-exceptional Cartier divisor $F$ on $Y$. Hence well done. $\blacksquare$

There is a generalization (known as Kawamata–Viehweg vanishing theorem) of this theorem, we refer section 3.2 in [Fujino17]⁸. $\blacksquare$

1. [Convex97] Ralph Tyrell Rockafellar. Convex Analysis. Princeton University Press. 1997. ↩

2. [SMMP] Singularities of the Minimal Model Program. Janos Kollar. Cambridge. 2013. ↩ ↩²

3. [Har77] Robin Hartshorne. Algebraic geometry, volume 52. Springer Science & Business Media, 1977. ↩

4. [FulIT2nd] William Fulton. Intersection Theory, 2nd. Springer New York, NY. 1998. ↩

5. [Positivity-I] R. Lazarsfeld. Positivity in AG I. Springer. 2000. ↩ ↩² ↩³

6. [GD] KARL SCHWEDE. GeneralizedDivisors. ↩ ↩²

7. [Sza95] Endre SzabÃ. Divisorial log terminal singularities. J. Math. Sci. Univ. Tokyo. Vol. 1 (1995), No. 3,631-639. ↩

8. [Fujino17] Osamu Fujino. Foundations of the Minimal Model Program. World Scientific. 2017. ↩ ↩² ↩³ ↩⁴

• algebraic-geometry